Problem: Find the distance between the point ${(5, 3)}$ and the line $\enspace {x = -2}\thinspace$. ${1}$ ${2}$ ${3}$ ${4}$ ${5}$ ${6}$ ${7}$ ${8}$ ${9}$ ${\llap{-}2}$ ${\llap{-}3}$ ${\llap{-}4}$ ${\llap{-}5}$ ${\llap{-}6}$ ${\llap{-}7}$ ${\llap{-}8}$ ${\llap{-}9}$ ${1}$ ${2}$ ${3}$ ${4}$ ${5}$ ${6}$ ${7}$ ${8}$ ${9}$ ${\llap{-}2}$ ${\llap{-}3}$ ${\llap{-}4}$ ${\llap{-}5}$ ${\llap{-}6}$ ${\llap{-}7}$ ${\llap{-}8}$ ${\llap{-}9}$
Solution: First, find the equation of the perpendicular line that passes through ${(5, 3)}$ Since the blue line has an infinite slope, the perpendicular line will have a slope of ${0}$ and therefore will be a horizontal line. The equation of the perpendicular line that passes through ${(5, 3)}$ is $\enspace {y = 3}\thinspace$ We can see from the graph that the two lines intersect at the point ${(-2, 3)}$ . Thus, the distance we're looking for is the distance between the two red points. Since their $y$ components are the same, the distance between the two points is simply the change in $x$ $|{5} - ( {-2} )| = 7$ The distance between the point ${(5, 3)}$ and the line $\enspace {x = -2}\enspace$ is $\thinspace7$.